Discrete valuation ( $v : K^{\times} \to Z$ )

A discrete valuation on a field

K

is a surjective group homomorphism

v : K^{\times} \to Z

satisfying

v (x + y) \geq min (v (x), v (y))

whenever

x + y \neq = 0

. Its valuation ring is

{0} \cup {x : v (x) \geq 0}

Table of contents
- Discretely valued field Discrete valuation
- Value group Discrete valuation

Discretely valued field

 0  0

A discretely valued field is a field equipped with a discrete valuation. It is complete when every Cauchy sequence for the induced metric converges.

Value group ( $v (K^{\times})$ )

 0  0

The value group of a valued field is the ordered abelian group formed by the values of its nonzero elements. A valuation is discrete when its value group is infinite cyclic with the order inherited from

Z

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A **discrete valuation** is a special type of valuation defined on a field, which gives a way to measure the "size" of elements in that field. More specifically, a discrete valuation provides a way to assess how "close" elements are to zero in a field, often in the context of algebraic number theory or local fields.

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Discrete valuation (v:K×→Z)